# Time dependent perturbation for harmonic oscillator

1. Mar 19, 2012

### Vandmelon

1. The problem statement, all variables and given/known data
I'm looking at the 1d harmonic oscillator

V(x)=\frac{1}{2}kx^2

with eigenstates n and the time dependent perturbation

H'(t)=qx^3\frac{(\tau^2}{t^2+\tau^2}

For t=-∞ the oscillator is in the groundstate n=0
I need to show that for n>3 the states will not get excited and I only need to look at the first order perturbation.

3. The attempt at a solution
What I'm thinking is $$P_{a→b}=\vert c_b(t)\vert^2$$ therefore I need to find when c_b=0
and because

c_b'=-\frac{i}{\hbar}∫H_{ba}'(t')e^{i\omega_0t'}dt'

I need to find

H_{ba}'=<\psi_n\vert H' \vert \psi_a >=0

For n>3
It would make sense if H_ba look a bit like this

H_{ba}'=\frac{d^n}{dx^n}H'

But I get some horrible results if I try to solve H_ba. Is it right what I am doing or is it way off.

2. Mar 19, 2012

### fzero

You want to compute $\langle n| H' |0\rangle$, which is proportional to $\langle n| \hat{x}^3 |0\rangle$. If you express $\hat{x}$ in terms of raising and lowering operators $\hat{a},\hat{a}^\dagger$, you should be able to see how to reach the necessary result.

3. Mar 19, 2012

### Vandmelon

so if I use

\hat x=\sqrt{\frac{\hbar}{2m\omega}}(a_+ + a_-)

then if I solved it right

\hat x^3 \psi_0=(\frac{\hbar}{2m\omega})^{\frac{3}{2}}[(a_+)^3+a_+ + a_- + (a_-)^3]\psi_0

Is it correct if I use a_- on ψ_0 it will just be 0 so

\hat x^3 \psi_0=(\frac{\hbar}{2m\omega})^{\frac{3}{2}}[(a_+)^3\psi_0+a_+\psi_0]=(\frac{\hbar}{2m\omega})^{\frac{3}{2}}[\sqrt6 \psi_3+\psi_1]

So it will end up like this

\langle \psi_n \vert\sqrt6 \psi_3+\psi_1 \rangle

If this is correct I am not sure why the oscillator wouldn't get excited for n>3. Or is it my calculations there are wrong?

*EDIT
Oh is it because

\langle \psi_n \vert \psi_m \rangle = 0

for n ≠ m

Last edited: Mar 19, 2012
4. Mar 19, 2012

### fzero

You'll want to go back through the algebra here. Remember that $a_+,a_-$ don't commute, so you have to keep track of the order of terms.

You will have to use both of these relations. You have all of the right ideas, just fix the algebra.